Problem 7 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 1 through 20 , find the indicated indefinite integral. \(\int\left(t^{5}-3 t^{2}+\frac{1}{t^{2}}\right) d t\)

Step-by-Step Solution

Answer

The indefinite integral of is.

Step 1 - Separate the Integral

Separate the given integral into individual integrals:ttttt.

Step 2 - Simplify Integrals

Rewrite the integrals for ease of integration. Specifically, t.

Step 3 - Integrate Each Term

Integrate each term separately. For the first term, use the power rule:t,t,For the second term, again use the power rule:t.The third term requires rewriting before using the power rule:t,t

Step 4 - Combine the Integrals and Add Constant

where C is the constant of integration.

Key Concepts

Power Rule in Integration

When solving indefinite integrals, the power rule in integration is a fundamental technique. The power rule helps to find antiderivatives of polynomial expressions. For any function of the form \(x^n\), the integral is found using:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Here, \(n\) is any real number except \(-1\), and \(C\) is the constant of integration.

• For example, integrating \(t^5\):
  \[ \int t^5 \, dt = \frac{t^{6}}{6} + C \], you increase the exponent by 1 and then divide by the new exponent.
  
• Similarly, \[ \int t^{-2} \, dt = \frac{t^{-1}}{-1} + C = -\frac{1}{t} + C \]
  This makes it clear and easy to integrate.
  

Integration of Polynomials

Polynomials are expressions made up of variables raised to non-negative integer powers and coefficients. To integrate polynomial expressions term by term, apply the power rule to each term individually.

• For instance, given the integral: \[ \int(t^5 - 3t^2 + \frac{1}{t^2}) \, dt \]

Split the integral into separate terms:
\[ \int t^5 \, dt - 3 \int t^2 \, dt + \int \frac{1}{t^2} \, dt \]

• Next, rewrite the integrals in a simpler form, especially dealing with fractions by writing \(1/t^2\) as \(t^{-2}\). This gives: \[ \int t^5 \, dt - 3 \int t^2 \, dt + \int t^{-2} \, dt \]

Then integrate each term separately:
\[ \int t^5 \, dt = \frac{t^6}{6} + C_1 \]
\[ -3 \int t^2 \, dt = -3 \frac{t^3}{3} + C_2 \]
\[ \int t^{-2} \, dt = -\frac{1}{t} + C_3 \]

Combining these results gives the final integrated expression:
\[ \frac{t^6}{6} - t^3 - \frac{1}{t} + C \] where \(C = C_1 + C_2 + C_3\) is the combined constant of integration.

Constant of Integration

In indefinite integration, the constant of integration \(C\) plays a vital role. After finding the antiderivative of a function, the constant of integration \(C\) is added to represent the entire family of antiderivative functions. This is important because:

• The process of differentiation of any constant is zero. Thus, differentiating \$f(x) + C$$ returns \$f'(x)$$.
  
• It ensures that all possible shifts of the antiderivative function are included, covering all solutions.
  

For example, consider the integral:
\[ \int t^5 \, dt = \frac{t^6}{6} + C \]
Here, \(C\) is an arbitrary constant. Even if different initial conditions or additional data are provided, \(C\) will adjust to match those conditions. This flexibility is crucial in applications, such as solving differential equations, where initial or boundary conditions need to be satisfied.

• Remember that without \(C\), the solution is incomplete. Always include the constant of integration in indefinite integrals.